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The Dolbeault operator on Hermitian spin surfaces. (English) Zbl 0987.53011
Authors’ summary: We consider the Dolbeault operator $$\sqrt 2(\overline\partial+ \overline\partial^*)$$ of $$K^{{1\over 2}}$$ – the square root of the canonical line bundle which determines the spin structure of a compact Hermitian spin surface $$(M,g,J)$$. We prove that all cohomology groups $$H^i(M,{\mathcal O}(K^{{1\over 2}}))$$ vanish if the scalar curvature of $$g$$ is nonnegative and non-identically zero. As a consequence, zero is not an eigenvalue of the Dolbeault operator. Moreover, we estimate the first eigenvalue of the Dolbeault operator when the conformal scalar curvature $$k$$ is nonnegative and when $$k$$ is positive. In the first case we give a complete list of limiting manifolds and in the second we give non-Kähler examples of limiting manifolds.

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53B35 Local differential geometry of Hermitian and Kählerian structures
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